Hubbard square 8x8, periodic/antiperiodic, U = 6, n = 1

Hubbard/square_64_PA_32_6

Record

-0.6574(2) E/site

VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16, fully parametrized hidden sub-matrix, hidden-unit density alpha = 1)

Moreno et al. (2022) · lowest eligible strict variational bound (rules §6)

The Hamiltonian

H = −t ∑_⟨ij⟩,σ (c^†_iσc_jσ + h.c.) + U ∑_i n_i↑n_i↓

t = 1, U = 6, N_↑ = N_↓ = 32 on 64 sites.

lattice: square
sites: 64
boundary: periodic/antiperiodic
Nf: 32
U: 6
energies: 3

Stored as a total energy in the convention of VarBench; quoted here as E/site, which is what the papers report (how the conversion works).

Every published energy

Grouped by what the number is. Ranking happens only inside the first group.

Strict variational upper bounds (2)

The record is the lowest eligible energy in this group, and only this group.

E/site	σ	Var(E)	V-score	method	source	year
-0.6668(1)	1.1e-4	8.20e-1	2.7e-3	VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16. Single hidden layer fully connected net with alpha = 1). Soft mean-field constraint for Neel order. flagged: below-exact	Moreno et al. (2022)	2022
-0.6574(2)record	2.0e-4	n/a	n/a	VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16, fully parametrized hidden sub-matrix, hidden-unit density alpha = 1) peer reviewed	Moreno et al. (2022)	2022

Numerically exact (1)

Exact diagonalization, an exact solution, or sign-problem-free QMC where that is established: the answer, not a claim about it.

E/site	σ	Var(E)	V-score	method	source	year
-0.6589(3)	3.1e-4	n/a	n/a	AFQMC, sign-problem-free at half filling (tau = 0.01, Trotter error below the statistical error) peer reviewed	Qin, Shi & Zhang, Benchmark stud	n/a

Flagged rows

A flag withholds the record and nothing else: the row stays listed, in rank order. Rules §10 is how a flag is lifted or upheld.

below-exact — VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16...., -0.6668(1)

Finding. VarBench stores -42.676 (E/N = -0.666813) and cites the HFDS paper, Moreno et al., PNAS 119, e2122059119 (2022). The paper gives -0.6574(2) for this lattice (8x8, half filling, U = 6, periodic along one side and antiperiodic along the other) in both the PNAS supplement (Table 5, read via Europe PMC PMC9371695) and the arXiv version (SI Table V). The stored number also sits 0.5 below the sign-free AFQMC total for this lattice, -42.17(2) (Qin, Shi & Zhang, PRB 94, 085103 (2016), Table IV) - 1.2%, about 25 sigma - which no variational energy can do. It breaks the paper's own size trend at U = 6 as well (-0.68135, -0.6609, -0.6574 for L = 4, 6, 8).

Diagnosis. The published value is a minimum of the optimization trace, not a converged measurement. Three seeds each, netket 3.22.4, alpha=1 complex RBM, 2000 SR steps. Evaluating the trained parameters by FULL SUMMATION over all 1024 basis states (zero Monte-Carlo error) puts every converged energy ABOVE the exact value, as the variational principle requires, while every training trace dips 1.5e-3 to 3.5e-3 BELOW it. Each published value lies between the two.

Ruled out. Not an autocorrelation-underestimated error bar: tau_corr <= 0.05 and R_hat = 1.0000 across all six runs.

Evidence. checks/results-tfising-rbm.json, checks/tfising_rbm_check.py

How these numbers were read

Rows added after the VarBench import record where the number came from.

-0.6589(3) — AFQMC, sign-problem-free at half filling (tau = 0.01...

Checked 2026-09-15 · arXiv PDF extracted locally with pypdf in layout mode, E rows of Table IV parsed by column position; no LLM transcription

Reported as. E = -42.17(2) (total), 8 x 8, U = 6, PBC-APBC

Table IV, "Total ground state energy (E) ... in the Hubbard model at half-filling, for U = 2, 4, 6, 8 ... for both PBC and PBC-APBC. Statistical errors are on the last digit." Row 8 x 8, column U = 6 PBC-APBC. Half filling: N_up = N_dn = 32. Cross-checks: the 8 x 8 PBC values at U = 4 and 8 agree with VarBench's own AFQMC exact rows to 1.3 sigma, and the 8 x 8, 10 x 10 and 12 x 12 PBC U = 8 values with the AFQMC references quoted by arXiv:2507.02644 (Table S1) to the printed digits.

-0.6574(2) — VMC Hidden Fermion Determinant State Ansatz (N_hidden = 16...

Checked 2026-09-15 · PNAS supplementary PDF (via Europe PMC, PMC9371695) extracted locally, cross-read against the arXiv version; no LLM transcription

Reported as. -0.6574(2) per site, no variance

PNAS SI Table 5 (identical to arXiv SI Table V), "Variational energy per site in the L x L Hubbard model at half filling with periodic boundary conditions along one of the sides of the square and anti-periodic boundary conditions along the other side", row L = 8, column U = 6. SI Sec. 6: N_hidden = 16, alpha = 1 for 8x8, the settings in the VarBench method string. Column alignment checked as for square_36_PA_18_2. As a total, -42.0736 sits 2.3e-3 (relative) above the sign-free AFQMC value -42.17(2) for this lattice (Qin, Shi & Zhang, PRB 94, 085103 (2016), Table IV, PBC-APBC), as a variational energy must, and fits the paper's size trend at U = 6 (-0.68135, -0.6609, -0.6574 for L = 4, 6, 8).

Coverage

Newest published number here: 2022. 2 rows added from the 2026-09 literature sweep, on top of the VarBench snapshot of 2024-10-22.

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